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April  2011, 29(2): 485-503. doi: 10.3934/dcds.2011.29.485

A general theorem on necessary conditions in optimal control

1. 

Université de Lyon, Institut Camille Jordan, CNRS UMR 5208, 69622 Villeurbanne, France

Received  September 2009 Revised  March 2010 Published  October 2010

We give a relatively short and self-contained proof of a theorem that asserts necessary conditions for a general optimal control problem. It has been shown that this theorem, which is simple to state, provides a powerful template from which necessary conditions for various other problems in dynamic optimization can be directly derived, at the level of the state of the art. These include various extensions of the Pontryagin maximum principle and the multiplier rule.
Citation: Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485
References:
[1]

F. H. Clarke, Optimal solutions to differential inclusions,, J. Optim. Th. Appl., 19 (1976), 469. doi: doi:10.1007/BF00941488. Google Scholar

[2]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley-Interscience, 5 (1983). Google Scholar

[3]

F. H. Clarke, Necessary conditions in dynamic optimization,, Memoirs of the Amer. Math. Soc., 173 (2005). Google Scholar

[4]

F. H. Clarke, Necessary conditions in optimal control and in the calculus of variations,, In, (2008), 143. doi: doi:10.1007/978-3-7643-8482-1_11. Google Scholar

[5]

F. H. Clarke and M. R. de Pinho, The nonsmooth maximum principle,, Control and Cybernetics, (). Google Scholar

[6]

F. H. Clarke and M. R. de Pinho, Optimal control problems with mixed constraints,, SIAM J. Control Optim., (). Google Scholar

[7]

F. H. Clarke, Yu. Ledyaev and M. R. de Pinho, An extension of the Schwarzkopf multiplier rule in optimal control,, preprint, (2009). Google Scholar

[8]

F. H. Clarke and Yu. S. Ledyaev, Mean value inequalities in Hilbert space,, Trans. Amer. Math. Soc., 344 (1994), 307. doi: doi:10.2307/2154718. Google Scholar

[9]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998). Google Scholar

[10]

M. R. Hestenes, "Calculus of Variations and Optimal Control Theory,", Wiley, (1966). Google Scholar

[11]

A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control,", American Math. Soc., (1998). Google Scholar

[12]

L. W. Neustadt, "Optimization,", Princeton University Press, (1976). Google Scholar

[13]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, "The Mathematical Theory of Optimal Processes,", Wiley-Interscience, (1962). Google Scholar

[14]

R. B. Vinter, "Optimal Control,", Birkhäuser, (2000). Google Scholar

show all references

References:
[1]

F. H. Clarke, Optimal solutions to differential inclusions,, J. Optim. Th. Appl., 19 (1976), 469. doi: doi:10.1007/BF00941488. Google Scholar

[2]

F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley-Interscience, 5 (1983). Google Scholar

[3]

F. H. Clarke, Necessary conditions in dynamic optimization,, Memoirs of the Amer. Math. Soc., 173 (2005). Google Scholar

[4]

F. H. Clarke, Necessary conditions in optimal control and in the calculus of variations,, In, (2008), 143. doi: doi:10.1007/978-3-7643-8482-1_11. Google Scholar

[5]

F. H. Clarke and M. R. de Pinho, The nonsmooth maximum principle,, Control and Cybernetics, (). Google Scholar

[6]

F. H. Clarke and M. R. de Pinho, Optimal control problems with mixed constraints,, SIAM J. Control Optim., (). Google Scholar

[7]

F. H. Clarke, Yu. Ledyaev and M. R. de Pinho, An extension of the Schwarzkopf multiplier rule in optimal control,, preprint, (2009). Google Scholar

[8]

F. H. Clarke and Yu. S. Ledyaev, Mean value inequalities in Hilbert space,, Trans. Amer. Math. Soc., 344 (1994), 307. doi: doi:10.2307/2154718. Google Scholar

[9]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, "Nonsmooth Analysis and Control Theory,", Graduate Texts in Mathematics, 178 (1998). Google Scholar

[10]

M. R. Hestenes, "Calculus of Variations and Optimal Control Theory,", Wiley, (1966). Google Scholar

[11]

A. A. Milyutin and N. P. Osmolovskii, "Calculus of Variations and Optimal Control,", American Math. Soc., (1998). Google Scholar

[12]

L. W. Neustadt, "Optimization,", Princeton University Press, (1976). Google Scholar

[13]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mischenko, "The Mathematical Theory of Optimal Processes,", Wiley-Interscience, (1962). Google Scholar

[14]

R. B. Vinter, "Optimal Control,", Birkhäuser, (2000). Google Scholar

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